Is the person responsible for the institution liable for the lost sum resulting from a fraud operation the institution was exposed to, or is the institution itself liable?

If there was no negligence or dereliction on your part concerning what you were entrusted with, then you are not liable for what was lost from the institution due to theft, because a trustee is not liable unless he transgresses or is negligent concerning what he was entrusted with. This is based on the saying of the Prophet, peace be upon him: "There is no liability on a trustee."